In order to obtain a comparison with analytical data, two spherical multilayer structures are studied. The first is a layered dielectric sphere consisting of four layers of thickness \(0.25 a\)  with relative permittivities of \(\varepsilon _r̰ = \left\lbrace 3, 5, 8, 2\right\rbrace \)  (inner layer to outer layer). The second structure contains layers with relative permittivities \(\varepsilon _r̰ = \left\lbrace 1, 5, 1, 2\right\rbrace \)  and relative permeabilities \(\mu _r̰ = \left\lbrace 3, 1, 8, 1\right\rbrace \) . Eigenangles \(\alpha _n\)  of tracked modes for the dielectric and dielectric / magnetic spheres are depicted in Figs. REF  and REF , respectively. The dashed lines correspond to analytical Mie series solutionsThe diagonal transition matrix of layered spheres may be constructed analytically using the Mie series [1]} and this matrix may, in turn, be used to obtain characteristic values via (REF )., solid lines are from FDTD simulations in CST Studio Suite, and the markers are from FEM simulation in Altair FEKO. The square markers at \(ka \approx 3.54\)  were evaluated 10 % off the position of the equidistant frequency point (\(ka \approx 3.55\) ) due to the instability of FEKO's solver at the original frequency.
